#27350429

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17430

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator, prime radical	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1470	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1470	D₅₃
hypocenter	C₅₃	#17430
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1470

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(70,73,75,74)(71,77,79,78)(72,80,81,82)(76,83,84,85);(70,72,75,81)(71,76,79,84)(73,82,74,80)(77,85,78,83);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56,58,64)(55,59,60,62)(57,65,66,63)(61,69,68,67);(54,55,57,61,58,60,66,68)(56,62,63,69,64,59,65,67)(70,71,74,78,75,79,73,77)(72,76,80,83,81,84,82,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=gh,d^2=gh,f^2=h,b^a=be,e^a=eg,f^b=fg,i^b=i^52,d^c=dgh,i^d=i^52

Conjugacy classes & character table

887 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135905);